Graphs and Matroids Weighted in a Bounded Incline Algebra

Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied.


Introduction
In graph theory, the famous shortest path problem (SPP, for short) is the problem of finding a path between two vertices in a weighted graph such that the sum of the weights of its constituent edges is minimized [1]. An example is finding the quickest way to get from one location to another on a road map. In this case, the vertices represent locations and the edges represent segments of road and are weighted by the time needed to travel that segment.
If we assume the weighted function to be nonnegative, then the related algebraic foundation of SPP is the semiring ([0, +∞], min, +). Therein, we use the operation "+" to compute the length of paths and use the operation "min" to find the least one. For the widest path problem (WPP, for short) or called the greatest capacity problem (GCP, for short), the related algebraic foundation is the semiring ([0, +∞], max, min). Accordingly, we use the operation "min" to compute the capacities and use the operation "max" to find the greatest one. For the most reliable path problem (MRPP, for short), the related algebraic foundation is the semiring ([0, 1], max, ×). Accordingly, we use the operation "×" to compute the reliability of paths and use the operation "max" to find the greatest one. There are many other classical problems using various semirings in graph theory [2].
For both ([0, +∞], min, +) for SPP and ([0, +∞], max, min) for WPP as well as the corresponding algorithms, the value "+∞" is used to act as the weight of artificial edges between vertex pairs with no edge. For these reasons, SPP, WPP, and MRPP (and other potential problems) can be put into a more generalized setting: the algebraic path problem [2]. The first aim of this paper is to unify SPP, WPP, WPP, and other path problems into graphs weighted in an idempotent semiring (also known as a dioid) [3]. We shall give a unified approach to find the shortest path, the widest path, and the most reliable path as well as their length.
In 1935, Whitney introduced matroids as a generalization of both graphs and linear independence in vector spaces [4]. It is well known that matroids play an important role in applied mathematics, especially in optimal theory, which are precisely the structures for the maximum independent set problem (MISP, for short) which the very simple and efficient greedy algorithm works [5]. The second aim of this paper is to study matroids weighted in a linear dioid and the related MISP.

Semirings, Incline Algebras, Dioids, and Their Properties
Semirings and matrices over semirings are useful tools in diverse areas such as automata theory, design of switching circuits, graph theory, medical diagnosis, Markov chains, 2 The Scientific World Journal informational systems, complex systems modeling, decisionmaking theory, dynamical programming, control theory, nervous system, probable reasoning, psychological measurement, and clustering [3,6].
In every dioid ( , ⊕, ⊗, 0, 1), we define ⪯ iff ⊕ = . Then ⪯ is a partial order on and ( , ⊕) is a bounded join semilattice. Clearly, 0 is the bottom element and 1 is the top element, so is the name bounded incline.
Similarly, ⊗ ⪯ ⊗ . Hence ⊗ ⪯ ⊗ . is a dioid, which is an algebraic model for SPP. The partial order ⪯ defined above is dual to the usual one ≤. For explicit, ⪯ iff ≤ in usual meaning.

Graphs Weighted in a Dioid and the Longest Path Problem
For the dioid ([0, +∞], min, +, +∞, 0) in SPP, since the partial order ⪯ is dual to the usual one ≤, the SPP in the dioid situation comes to be a longest path problem (LPP for short).
In this section, we will study the LPP for graphs weighted in The Scientific World Journal 3 a dioid, which can be considered a unified approach for SPP, WPP, and MRPP (and so on).
Let be a graph weighted in a dioid ( , ⊕, ⊗, 0, 1). For two vertices , , let denote the set of paths from to . For ∈ , = ⊗ ∈ ( ) is called the length of the path . Since ⊕ is idempotent, = ⊕ ∈ is the longest path length from V to V . For the weighted graph , define a matrix = ( ) × by the following: (1) for ̸ = , if there are some paths from V to V , then put as the maximal weight of all parallel edges from V to V ; if there is no path from V to V , then put = 0; (2) for every , put = 1.  (1) ). Let 1 , 2 , . . . , , +1 be an ( + 1)-step path from V to V . Then 1 , 2 , . . . , is an -step path from V to V , where V is the end vertex of ; and +1 is an edge from V to V and (1) ⪰ ( +1 ). Then . This completes the proof.
(3) Suppose that ≥ . By (2), we have ( ) ⪰ ( ) . By (1), ( ) is the longest -step path from V to V . Suppose that the related path of ( ) is = {V 1 , V 2 , . . . , V }, since there is at most vertices in and there are some common points in . Suppose that V and V (let ≤ ) are the same point. In order to make ( ) the longest path, it must hold that V = V = V for all ≤ ≤ . Then the length ( ) is equal to the length of a path from V to V with no common point (with at most vertices). Hence ( ) ⪯ ( ) since ( ) is the longest -step path from V to V .
Similar to weighted graph, matroids also play an important role in mathematics, especially in applied mathematics, which are precisely the structures for which the very simple and efficient greedy algorithm works [5].
In this section, we will study matroids weighted in a linear dioid (notice that all the examples in Examples 5 and 6 are linear) and the maximum independent set problem.
We suppose that ( , ⊕, ⊗, 1, 0) is a linear dioid. Let be a finite set and = ( , I) a matroid weighted in with : → being the weighted function.
In the optimization theory, the maximum independent set problem (MISP, for short) is to find an independent subset ∈ I( ) such that ( ) = max{ ( ) | ∈ I( )}. We will use the famous greedy algorithm to deal with this problem.
Proposition 11. The greedy algorithm has an optimal solution.
Proof. Suppose that is a solution of GA. Then ∈ I( ).